Topologically Protected Complete Polarization Conversion

We consider the process of conversion between linear polarizations as light is reflected from a photonic crystal slab. We observe that, over a wide range of frequencies, complete polarization conversion can be found at isolated wave vectors. Moreover, such an effect is topological: the complex reflection coefficients have a nonzero winding number in the wave vector space. We also show that bound states in continuum in this system have their wave vectors lying on the critical coupling curve that defines the condition for complete polarization conversion. Our work points to the use of topological photonics concepts for the control of polarization, and suggests the exploration of topological properties of scattering matrices as a route towards creating robust optical devices.

Figure 1

(a) Schematic of the structure. A dielectric photonic crystal is sitting on top of a perfect mirror. We denote the dielectric constant of the slab by $\epsilon$, the periodicity by $a$, the thickness of the slab by $h$, the radius of the air hole by $r$. For the other subplots, the parameters are $\epsilon =12$, $h=0.3a$, $r=0.3a$. (b) Blue curve: reflection spectrum at $k_{\parallel ,1}=(0.0522,0.0240)2\pi /a$. Red curve: reflection spectrum at $k_{\parallel ,2}=(0.0717,0.0345)2\pi /a$. The complete conversion where $R_{ss}=0$ occurs at ${\omega }_{0,1}=0.388\times 2\pi c/a$ and ${\omega }_{0,2}=0.385\times 2\pi c/a$, respectively. (c), (d) $|R_{ss}|$ at the frequency (c) ${\omega }_{0,1}$, (d) ${\omega }_{0,2}$. (e), (f) Direction flow of vector field $\mathbf{(}\Re (R_{ss}),\Im (R_{ss})\mathbf{)}$ at the frequency (e) ${\omega }_{0,1}$, (f) ${\omega }_{0,2}$.

Figure 2

(a) Band diagram of the structure in Fig. 1 with $\epsilon =12$, $h=0.3a$, $r=0.3a$. The dashed lines denote the light cone. (b), (c), (d) Electric field ($\Re (E_z)$) distribution of the lowest order guided resonance at the interface between photonic crystal slab and mirror at (b) $\mathrm{\Gamma }$ point, (c) $(k_x,k_y)=(0.1,0.0)2\pi /a$ on $\mathrm{\Gamma }X$, (d) $(k_x,k_y)=(0.1,0.1)2\pi /a$ on $\mathrm{\Gamma }M$, where red color denotes positive values and blue color denotes negative values.

Figure 3

The parameters are $\epsilon =12$, $h=0.3a$, $r=0.3a$, which are the same as those in Fig. 1 and Fig. 2. (a) Amplitude of coupling constant to $s$-polarized waves of the lowest band, denoted by $|d^s|$. $|d^s|$ is zero along $\mathrm{\Gamma }X$. (b) Amplitude of the coupling constant to $p$-polarized waves of the lowest band, denoted by $|d^p|$. $|d^p|$ is zero along $\mathrm{\Gamma }M$. (c) The background shows quality factor of the lowest order resonances in a logarithmic scale. The dashed curve denotes the critical coupling curve on which $|d^s|=|d^p|$.

Figure 4

The parameters here are $\epsilon =4$, $h=0.9a$, $r=0.4a$. (a) Band diagram of the structure. The dashed lines enclose the region where only zeroth order diffraction can occur. The two blue circles indicate guided resonances with infinite quality factor. (b) The background shows quality factor of the lowest order resonances in a logarithmic scale. The dashed curve denotes the critical coupling curve on which $|d^s|=|d^p|$. The blue circles indicate the wave vectors of the non-symmetry-protected bound states in continuum where $|d^s|=|d^p|=0$. (c) Amplitudes of coupling constants to $s$-polarized waves of the lowest band, denoted by $|d^s|$. The blue circle indicates the wave vector on $\mathrm{\Gamma }M$ for which $|d^s|=0$. (d) Amplitude of coupling constant to $p$-polarized waves of the lowest band, denoted by $|d^p|$. The blue circle indicates the wave vector on $\mathrm{\Gamma }X$ for which $|d^p|=0$.